Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3234,2,Mod(1,3234)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3234.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3234, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,-1,1,0,1,0,-1,1,0,1,-1,4,0,0,1,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.8236200137\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3234.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} -1.00000 q^{22} -3.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} -4.00000 q^{26} -1.00000 q^{27} -9.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} -7.00000 q^{37} -1.00000 q^{38} -4.00000 q^{39} +6.00000 q^{41} +11.0000 q^{43} +1.00000 q^{44} +3.00000 q^{46} +3.00000 q^{47} -1.00000 q^{48} +5.00000 q^{50} +3.00000 q^{51} +4.00000 q^{52} +1.00000 q^{54} -1.00000 q^{57} +9.00000 q^{58} -9.00000 q^{59} +10.0000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +1.00000 q^{66} -4.00000 q^{67} -3.00000 q^{68} +3.00000 q^{69} +3.00000 q^{71} -1.00000 q^{72} +4.00000 q^{73} +7.00000 q^{74} +5.00000 q^{75} +1.00000 q^{76} +4.00000 q^{78} -16.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -11.0000 q^{86} +9.00000 q^{87} -1.00000 q^{88} -3.00000 q^{92} +2.00000 q^{93} -3.00000 q^{94} +1.00000 q^{96} +1.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416 0.114708 − 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −3.00000 −0.625543 −0.312772 − 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 1.00000 0.204124
\(25\) −5.00000 −1.00000
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −1.00000 −0.162221
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 11.0000 1.67748 0.838742 − 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 3.00000 0.437595 0.218797 − 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 5.00000 0.707107
\(51\) 3.00000 0.420084
\(52\) 4.00000 0.554700
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 9.00000 1.18176
\(59\) −9.00000 −1.17170 −0.585850 − 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −3.00000 −0.363803
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 − 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) −1.00000 −0.117851
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 7.00000 0.813733
\(75\) 5.00000 0.577350
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) −16.0000 −1.80014 −0.900070 − 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −11.0000 −1.18616
\(87\) 9.00000 0.964901
\(88\) −1.00000 −0.106600
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.00000 −0.312772
\(93\) 2.00000 0.207390
\(94\) −3.00000 −0.309426
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 1.00000 0.101535 0.0507673 − 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3234.2.a.f.1.1 1
3.2 odd 2 9702.2.a.bn.1.1 1
7.3 odd 6 462.2.i.b.331.1 yes 2
7.5 odd 6 462.2.i.b.67.1 ✓ 2
7.6 odd 2 3234.2.a.l.1.1 1
21.5 even 6 1386.2.k.d.991.1 2
21.17 even 6 1386.2.k.d.793.1 2
21.20 even 2 9702.2.a.bl.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.b.67.1 ✓ 2 7.5 odd 6
462.2.i.b.331.1 yes 2 7.3 odd 6
1386.2.k.d.793.1 2 21.17 even 6
1386.2.k.d.991.1 2 21.5 even 6
3234.2.a.f.1.1 1 1.1 even 1 trivial
3234.2.a.l.1.1 1 7.6 odd 2
9702.2.a.bl.1.1 1 21.20 even 2
9702.2.a.bn.1.1 1 3.2 odd 2